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Claim: if $f$ is a function and $g$ is a function and $a$ is the only solution to equation $$g(x)=g(a)$$ and $$\lim_{x\to a} g(x) = g(a)$$ and $$\lim_{y \to g(a)} f(y) = L$$ then $$\lim_{x\to a} f(g(x)) = L$$

Proof: $$\lim_{x\to a} g(x) = g(a)$$ so for every positive real number $\epsilon$ there exists positive real number $\delta$ such that from the fact $$0<|x-a|<\delta$$ follows $$|g(x)-g(a)|<\epsilon$$
Similiraly, $$\lim_{y\to g(a)} f(y) = L$$ so for every positive real number $\zeta$ there exists positive real number $\Delta$ such that from the fact $$0<|y-g(a)|<\zeta$$ follows $$|f(y)- L|<\Delta$$ Now we set y = g(x).
From the first limit we have that there exists some $\delta$ that from $$0<|x-a|<\delta$$ we get $$|g(x)-g(a)|<\zeta$$ because from the first limit we know that $\epsilon$ is any real number. But we know that $a$ is the only solution to the equation $$g(x)=g(a)$$ so from the fact that $$0<|x-a|<\delta$$ we can conclude that $$0<|g(x)-g(a)|<\zeta$$ Now we have that from the fact that $$0<|x-a|<\delta$$ we conclude that $$0<|g(x)-g(a)|<\zeta$$ and from that we conclude that $$|f(g(x))- L|<\Delta$$ so
$$\lim_{x\to a} f(g(x)) = L$$ Q.E.D.
Is this proof valid ? If so what are other sufficient conditions for such a chain rule ?

Юрій Ярош
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    I believe most keyboards have a vertical bar on the backslash key used for LaTeX commands, it is usually shift + \ to get |, though abs(x) is still a perfectly good name for the function. Alternatively you could use \vert to get a vertical line though that's more work. – Triatticus Mar 08 '18 at 15:25
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    \lvert and \rvert are left and right delimiters, respectively. –  Mar 08 '18 at 15:28
  • Triatticus and Kevin thanks for advises. – Юрій Ярош Mar 08 '18 at 15:35
  • @Triaticus thanks for advises – Юрій Ярош Mar 08 '18 at 15:42
  • @Kevin thanks for advises – Юрій Ярош Mar 08 '18 at 15:42
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    "$\lim_{g(x)\to g(a)} f(g(x)) = L$" what does that mean? – zhw. Mar 08 '18 at 22:01
  • @zhw. It is a limit of a function $f$ with an input of $g(x)$ at point $g(a)$. – Юрій Ярош Mar 08 '18 at 22:04
  • @zhw. What confuses you in that limit ? The fact that we have limit as $g(x)$ approaches $g(a)$ ? If it is so we can set $y=g(x)$ and the proof would be the same. – Юрій Ярош Mar 08 '18 at 22:09
  • The limit notation involves variables and functions of variables. It has to be of the form $$\lim\limits_{\text{variables }\to\text{ values}} \text{ function of variables} $$ and not like $$\lim\limits_{\text{function }\to\text{ values}} \text{ function of variables}$$ So writing like $\lim_{g(x) \to g(a)} $ is non-standard. – Paramanand Singh Mar 09 '18 at 05:07
  • Also there is limit of function at a point, not that limit of function "with some input" at some point. In symbols the notation $y\to g(a) $ and $g(x) \to g(a) $ are different. In first case $y$ takes all values in a neighborhood of $g(a) $ and in second case not all values in neighborhood of $g(a) $ are taken by $g(x) $. – Paramanand Singh Mar 09 '18 at 05:10
  • For other sufficient conditions see https://math.stackexchange.com/a/1073047/72031 – Paramanand Singh Mar 09 '18 at 05:13
  • @ParamanandSingh I could just say that "it is a limit of a function $f$ at a point g(a)". Or it would change something ? – Юрій Ярош Mar 09 '18 at 09:36
  • Yes you should say that because that is the way to say it and just write $\lim_{x\to g(a)} f(x) =L$. You can also see my linked answer where I have given a result where the continuity of $g$ is not needed. – Paramanand Singh Mar 09 '18 at 09:39
  • @ParamanandSingh I saw it, but the problem is I am not familiar with "neighborhoods". – Юрій Ярош Mar 09 '18 at 09:49
  • @ParamanandSingh Yeah, it is nonstandard, but I don't see some restrictions for that in the limit definition, or am I wrong ? – Юрій Ярош Mar 09 '18 at 09:53
  • Neighborhood of a point $p$ is any open interval which contains $p$. Thus $(1,3)$ is a neighborhood of $2.95$. Further the notation and definition of limits is pretty standard and no textbook uses it in the manner you have used. It is better to use the standard notations to avoid ambiguity or otherwise you need to explain the non-standard notation. – Paramanand Singh Mar 09 '18 at 10:19
  • @ParamanandSingh Can't we just take standard definition of a limit and take $g(x)$ as an input of the function instead of $x$ and get what I wrote ? I mean, we have $$\lim_{x\to g(a)} f(x) = L$$ can't we just plug in $g(x)$ instead ? – Юрій Ярош Mar 09 '18 at 10:44
  • @ParamanandSingh Thanks for the definition of neighborhood, now I will try to understand your variant of "chain rule". – Юрій Ярош Mar 09 '18 at 10:46
  • Let us continue this discussion in chat. – Paramanand Singh Mar 09 '18 at 10:47
  • @ParamanandSingh OK, let's continue the discussion in the chat. – Юрій Ярош Mar 09 '18 at 10:51

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